The invention relates to Markov chain control of random modulated switching signals in a power converter.
Switching power converters are the most widespread systems in the area of power conditioning. The reasons for their popularity are numerous, the most outstanding being their ability to achieve a very high efficiency of operation, the capability to operate at different voltage and current levels, and the relative abundance of circuit topologies that can be matched to various requirements. The control of switching converters is an area of intensive growth. There exists an array of interesting control problems, motivated primarily by the wide range of operating conditions characterizing a power converter and the very constrained nature of control actions, for instance, one can only choose the instants at which power switches are closed or opened, to select one of a fairly small total number of circuit configurations. Many questions in power converter control are still not answered in sufficient detail (e.g. geometrical control, digital implementations), while a host of new questions arise naturally when the robustness of operation is considered.
The switching function for a given power switch, denoted by q(t), is a time waveform taking the value 1 when the switch is on, and the value 0 when the switch is off. In the case of DC/DC converters, the nominal switching function is typically periodic, with a period equaling the duration of a single on-off cycle, as shown in the switching function 10 of FIG. 1A. In the case of DC/AC converters,the nominal switching function is often periodic, with a period that comprises several on-off cycles, as indicated by the switching function 12 of FIG. 1B.
As shown in FIGS. 2A and 2B, a conventional switching process for a power converter 20 involves generating a switching function q(t) for a switching device 21 with a configuration including a controller 22, a clock 24, a comparator 26, and a latch 28. In this configuration, the reference values fed to the controller 22 reflect the desired steady-state quantities (e.g. voltages and currents). Any necessary feedback control signals are combined with these reference values to specify the modulating signal m(t), which in turn determines q(t). Since power converters generally operate in a periodic steady state, converter waveforms of interest are typically periodic functions of time in the steady state.
Converter waveforms which are periodic functions of time in general have spectral components at all integer multiples of the fundamental frequency. The allowable harmonic content of some of these waveforms is often constrained, an example is the current in the interface to the electric utility, when it is desirable to have only the 60 Hz fundamental component present. In this case, stringent filtering requirements may be imposed on the power converter operation. Since the filter size is in general related to these requirements, a significant part of a power converter's volume and weight can be due to an input or output filter. This conflicts with requirements to miniaturize power supply components, which have been the driving force behind much of modern power electronics.
Similar requirements hold for acoustic noise control in motor applications. Harmonic components of the motor voltages and currents may excite mechanical resonances, leading to increased acoustic noise and to possible torque pulsations. Present solutions to these problems include either a costly mechanical redesign, or an increase in the switching frequency in the power converter supplying the motor, which in turn increases the switching power losses.
In conventional random modulation processes, a signal with appropriately chosen statistical properties is added to the reference values utilized in the control configuration of FIG. 2A. This has the effect of randomly "dithering" q(t) from its nominal form. The randomization can alter the harmonic content of waveforms of interest without excessively affecting the proper operation of the converter. In terms of FIGS. 1A and 1B, randomization occurs in each cycle of the reference waveform.
As a common ground for comparisons among different random modulation methods is needed, it is useful to concentrate on the switching function q(t), which Can take only 0-1 values. Quantities of interest in a switching cycle are total cycle duration, duration of the on-portion of the cycle, and the position of the on-portion within the cycle. The ratio of the duration of the on-portion to the total cycle length is called the duty ratio. Many waveforms of interest in implementations are related to such a pulse train via linear transformations (e.g. a simple integral in the case of the input current of a boost converter). The power spectrum of variables related to q(t) by linear, time-invariant (LTI) operations can easily be derived from the power spectrum of q(t). The power spectrum of many other waveforms of interest can be derived by methods similar to those used for q(t).
The main elements characterizing a random modulation process are the time variation of the nominal (non-randomized) switching pattern and the time variation of the probability laws that govern the randomization. First, it is necessary to determine if the nominal patterns, e.g. duty ratios, vary from one cycle to the next, as they do in inverter operation. This property defines the deterministic structure of the modulation. The other issue is the time variation of the probability densities used to determine the "dither" at each cycle. This component is thought of as the probabilistic structure of the modulation.
If both the deterministic and probabilistic structures are constant in time (implying DC/DC operation), the switching will be called stationary. In block-stationary random modulation, the nominal pattern varies from cycle to cycle, but is repeated periodically over a block of cycles, as needed for inverter (DC/AC) operation. The present invention considers a third type of structure, where the probability density used for dither in each cycle depends on the state of a Markov chain at the beginning of that cycle. It will hereinafter be described that switching based on a Markov chain enables explicit control of the ripple, while maintaining analytical tractability.
Stationary switching processes can be further classified, and the most important classes are randomized pulse position modulation (PPM), randomized pulse width modulation (PWM), and asynchronous randomized modulation. FIG. 3 shows one cycle of the switching waveform; T.sub.i is the duration of the i-th cycle, a.sub.i is the on-time within a basic switching cycle, and .epsilon..sub.i is the position of the turn-on within the cycle. The duty ratio is d.sub.i =a.sub.i /T.sub.i. All switching functions q(t) that are analyzed in with respect to the present invention consist of concatenations of such switching cycles. In general, .epsilon..sub.i, d.sub.i or T.sub.i, can be dithered, individually or simultaneously. Some combinations used in power electronics are as follows:
Random PPM: .epsilon..sub.i changes; T.sub.i, a.sub.i fixed. PA1 Random PWM: a.sub.i changes; .epsilon..sub.i =0; T.sub.i fixed. Within random PWM, d.sub.i can be varied either continuously, or it can take finitely many distinct values. PA1 Asynchronous modulation: T.sub.i changes; .epsilon..sub.i =0; d.sub.i fixed. PA1 Simplified asynchronous modulation: T.sub.i is varied, a.sub.i is fixed, .epsilon..sub.i =0. PA1 Vary T.sub.i and d.sub.i simultaneously, .epsilon..sub.i =0, with predetermined time averages, PA1 Vary independently the "on" and "off" times, with predetermined averages. An example of this kind is the random telegraph wave with different transition rates from 0 to 1 and from 1 to 0.
Some other possibilities involve varying more than one variable simultaneously, or dithering their sums, differences and the like:
The main benefit of such processes in the case of converters supplying motors is acoustic noise reduction and torque pulsation reduction, and filter size reduction in all classes of power electronic converters. However, a possible practical drawback of conventional random modulation is the absence of a time-domain characterization. While the power spectrum of a waveforms of interest can now be accurately predicted, measured or estimated, there is no guarantee that the time domain waveform will not deviate arbitrarily from its desired average. This is a consequence of the commonly used random modulation procedure, which is based on statistically independent random experiments (trials).
According to the present invention, there is described a family of random modulation processes that are based on Markov chains that enable both deterministic and stochastic descriptions of time domain waveforms, in addition to the spectral shaping. Analytical formulas describing random modulation based on Markov chains are slightly more complicated than the corresponding formulas for the independent modulation case. These formulas, however, are used for switching strategy assessment and optimization (off-line), thus making the calculations entirely tractable.